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Collinear points
tenplusten 2
N
by Blackbeam999
Let 
be three collinear points and 
three other collinear points. Let 
be the intersection of the lines 
and 
,respectively. If 
and 
.Prove that 
I hope you will use Pappus theorem in your solutions.
be three collinear points and 
three other collinear points. Let 
be the intersection of the lines 
and 
,respectively. If 
and 
.Prove that 
I hope you will use Pappus theorem in your solutions.
2 replies
Simple Geometry
AbdulWaheed 0
Source: EGMO
Try to avoid Directed angles
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
Let ABC be an acute triangle inscribed in circle
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
0 replies
IMO Shortlist 2014 G6
hajimbrak 30
N
by awesomeming327.
Let be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. We call the pair 
, if the quadrilateral 
is cyclic.
Suppose that the pairs
and 
are interesting. Prove that 
Proposed by Ali Zamani, Iran
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. We call the pair 
, if the quadrilateral 
is cyclic.Suppose that the pairs
and 
are interesting. Prove that 
Proposed by Ali Zamani, Iran
30 replies
60^o angle wanted, equilateral on a square
parmenides51 4
N
by MathIQ.
Source: 2019 Austrian Mathematical Olympiad Junior Regional Competition , Problem 2
A square 
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
. Prove that 
.
.
(Karl Czakler)
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
. Prove that 
..
(Karl Czakler)
4 replies
Serbian selection contest for the IMO 2025 - P2
OgnjenTesic 9
N
by hectorleo123
Source: Serbian selection contest for the IMO 2025
Let be an acute triangle. Let 
be the reflection of point over the line . Let 
and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
Proposed by Strahinja Gvozdić
be an acute triangle. Let 
be the reflection of point over the line . Let 
and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.Proposed by Strahinja Gvozdić
9 replies
collinear wanted, regular hexagon
parmenides51 3
N
by MathIQ.
Source: 2023 Austrian Mathematical Olympiad , Junior Regional Competition , Problem 2
Let 
be a regular hexagon with sidelength s. The points 
and 
are on the diagonals 
and 
, respectively, such that 
. Prove that the three points 
, and are on a line.
(Walther Janous)
be a regular hexagon with sidelength s. The points 
and 
are on the diagonals 
and 
, respectively, such that 
. Prove that the three points 
, and are on a line.(Walther Janous)
3 replies
Geometric inequality with angles
Amir Hossein 7
N
by MathIQ.
Let 
, and 
be the angles of a triangle, and let 
, and 
. If 
, show that
![\[s(s - a)(s - b)(s -c) \geq 0.\]](//latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
When does equality hold?
, and 
be the angles of a triangle, and let 
, and 
. If 
, show that![\[s(s - a)(s - b)(s -c) \geq 0.\]](http://latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
When does equality hold?
7 replies
IMO 2014 Problem 3
v_Enhance 103
N
by Mysteriouxxx
Source: 0
Convex quadrilateral has 
. Point is the foot of the perpendicular from to . Points and lie on sides and 
, respectively, such that lies inside triangle 
and ![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](//latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)
Prove that line is tangent to the circumcircle of triangle 
.
has 
. Point is the foot of the perpendicular from to . Points and lie on sides and 
, respectively, such that lies inside triangle 
and ![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](http://latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)
Prove that line is tangent to the circumcircle of triangle 
.
103 replies
three discs of radius 1 cannot cover entirely a square surface of side 2
parmenides51 1
N
by Blast_S1
Source: 2014 Romania NMO VIII p4
Prove that three discs of radius 
cannot cover entirely a square surface of side 
, but they can cover more than 
of it.
cannot cover entirely a square surface of side 
, but they can cover more than 
of it.
1 reply
2025 Caucasus MO Juniors P6
BR1F1SZ 2
N
by IEatProblemsForBreakfast
Source: Caucasus MO
A point is chosen inside a convex quadrilateral . Could it happen that
is chosen inside a convex quadrilateral . Could it happen that
2 replies
